Theorem imaeq2 5019

Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Assertion

Ref	Expression
imaeq2	|- ( A = B -> ( C " A ) = ( C " B ) )

Proof of Theorem imaeq2

Step	Hyp	Ref	Expression
1		reseq2 4955	. . 3 |- ( A = B -> ( C |` A ) = ( C |` B ) )
2	1	rneqd 4908	. 2 |- ( A = B -> ran ( C |` A ) = ran ( C |` B ) )
3		df-ima 4689	. 2 |- ( C " A ) = ran ( C |` A )
4		df-ima 4689	. 2 |- ( C " B ) = ran ( C |` B )
5	2, 3, 4	3eqtr4g 2263	1 |- ( A = B -> ( C " A ) = ( C " B ) )

Syntax hints:   -> wi 4   = wceq 1373   ran crn 4677   |` cres 4678   "cima 4679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-xp 4682  df-cnv 4684  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689

This theorem is referenced by:  imaeq2i  5021  imaeq2d  5023  fimadmfo  5509  ssimaex  5642  ssimaexg  5643  isoselem  5891  f1opw2  6154  fopwdom  6935  ssenen  6950  fiintim  7030  fidcenumlemrk  7058  fidcenumlemr  7059  sbthlem2  7062  isbth  7071  ennnfonelemp1  12810  ennnfonelemnn0  12826  ctinfomlemom  12831  ctinfom  12832  tgcn  14713  iscnp4  14723  cnpnei  14724  cnima  14725  cnconst2  14738  cnrest2  14741  cnptoprest  14744  txcnp  14776  txcnmpt  14778  metcnp3  15016